Homogenization for locally periodic elliptic problems on a domain (2006.05856v2)
Abstract: Let $\Omega$ be a Lipschitz domain in $\mathbb Rd$, and let $\mathcal A\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $\Omega$. We suppose that $\varepsilon$ is small and the function $A$ is Lipschitz in the first variable and periodic in the second, so the coefficients of $\mathcal A\varepsilon$ are locally periodic and rapidly oscillate. Given $\mu$ in the resolvent set, we are interested in finding the rates of approximations, as $\varepsilon\to0$, for $(\mathcal A\varepsilon-\mu){-1}$ and $\nabla(\mathcal A\varepsilon-\mu){-1}$ in the operator topology on $L_p$ for suitable $p$. It is well-known that the rates depend on regularity of the effective operator $\mathcal A0$. We prove that if $(\mathcal A0-\mu){-1}$ and its adjoint are bounded from $L_p(\Omega)n$ to the Lipschitz--Besov space $\Lambda_p{1+s}(\Omega)n$ with $s\in(0,1]$, then the rates are, respectively, $\varepsilons$ and $\varepsilon{s/p}$. The results are applied to the Dirichlet, Neumann and mixed Dirichlet--Neumann problems for strongly elliptic operators with uniformly bounded and $\operatorname{VMO}$ coefficients.